Heat the oven to 400°F. Cut puff pastry into six equal rectangles, brush with the tablespoon milk and bake on a lined sheet for 12 to 15 minutes until puffed and golden.
Meanwhile, melt butter in a large soup pot and cook onion, carrots and celery for 6 minutes. Stir in flour for 1 minute, then gradually add broth.
Add potatoes, chicken, salt, thyme and pepper. Simmer gently for 15 to 18 minutes until the potatoes are tender and the chicken reaches 165°F.
Stir in peas and half-and-half and heat gently for 3 to 4 minutes until the peas are hot and the soup lightly coats a spoon.
Divide among bowls and place one baked pastry rectangle over each serving just before eating.